Physics in Nuclear Medicine

James A. Sorenson, Ph.D. Professor of Radiology Department of Radiology University of Utah Medical Center Salt Lake City, Utah

Michael E. Phelps, Ph.D. Professor of Radiological Sciences Department of Radiological Sciences Center for Health Sciences University of California Los Angeles, California

~ J Grune & Stratton A Subsidiary of Harcourt BraceJovanovich, Publishers New York London Paris San Diego San Francisco Sao Paulo Sydney Tokyo Toronto Radioactivity is a process involving events in individual atoms and nuclei. Before discussingradioactivity, therefore, it is worthwhile to review some of the basic conceptsof atomic and nuclearphysics.

A. MASS AND ENERGY UNITS

Events occurring on the atomic scale, such as radioactivedecay, involve massesand energiesmuch smaller than those encounteredin the eventsof our everydayexperiences. Therefore they are describedin termsof massand energy units more appropriateto the atomic scale. The basic unit of massis the universal massunit, abbreviatedu. One u is defined as being equal to exactly Y12the mass of a 12C atom. t A slightly different unit, commonly used in chemistry, is the atomic mass unit (amu), basedon the averageweight of oxygen isotopesin their natural abundance.In this text, except where indicated, masseswill be expressedin universal mass units, u. The basic unit of energy is the electron volt, abbreviatedeV. One eV is defined as the amount of energy acquiredby an electronwhen it is accelerated through an electrical potential of one volt. Basic multiples are the keV (kilo electron volt; 1 keY = 1000 eV) and the MeV (Mega electron volt; 1 MeV = 1000 keY = 1,000,000eV). Mass m and energy E are related to each other by Einstein's equationE = mc2,where c is the velocity of light. According to this equation, 1 u of mass is equivalentto 931.5 MeV of energy.

tAtomi~ notation is discussedin SectionD.2 of this chapter. 1 2 Physics in Nuclear Medicine

Table 1-1 Mass and Energy Units , I -.. Multiply - To Obtain To Obtain By 4i-- Divide

u 1.66043 X 10-24 g u 4.86 X 10-26 oz u 1.00083 amu eV 1.6021 X 10-12 ergs eV 3.83 X 10-20 g.cal u 931.478 MeV

Relationships between various units of mass and energy are summarized in Table 1-1. Universal mass units and electron volts are very small, yet, as we shall see, they are quite appropriate to the atomic scale.

B. ELECTROMAGNETIC RADIATION (PHOTONS)

Atomic and nuclear processes often result in the emission of electromagnetic radiation, such as x rays (Section C.3), and -y rays (gamma rays, Section D.5). Electromagnetic radiation consists of oscillating electrical and magnetic fields traveling through space with the velocity of light, c (approximately 3 X 108 meters/sec in vacuum). The wavelength A and frequency v of these oscillating fields are related by

AV = C (1-1)

In some cases, e.g., when interacting with atoms, electromagnetic radia- tions behave as discrete "packets" of energy, called photons (also called quanta). A photon is a packet of electromagnetic energy having no mass or electrical charge that also travels through space at the velocity of light. The energy of a photon, E, and the wavelength A of its associated electromagnetic field are related by

E(keV) = l2.4/A(A) (1-2)

A(A) = l2.4/E(keV) (1-3)

Table 1-2 lists approximate photon energies and wavelengths for different parts of the electromagnetic spectrum. Note that x and -y rays occupy the highest- energy, shortest-wavelength end of the spectrum; x- and -y-ray photons have energies in the keY-MeV range, whereas visible light photons, for example, have energies of only a few eV. As a consequence of their short wavelength Basic Atomic and Nuclear Physics 3

Table 1-2 ApproximatePhoton Energy and Wavelength Ranges for DifferentTypes of ElectromagneticRadiation Type Energy(eV) Wavelength Radio,TV, radar 10-9-10-3 10-I-I0~ cm Infrared 10-3-2 10-4-10-1cm Visible 2-3 4000-7000At Ultraviolet 3-25 50-4000A x rays 25-105 10-1-50A 'Yrays 104-106 10-2-1 A tl A = 10-8Cffi. and high energy, x and 'Y rays interact with matter quite differently from other, more familiar types of electromagneticradiation. These interactions are discussedin detail in Chapter9.

C. ATOMS

1. Composition and Structure All matter is comprisedof atoms. An atom is the smallestunit into which a chemical element can be broken down without losing its chemical identity. Atoms combine to form molecules and chemical compounds, which in turn combineto form larger, macroscopicstructures. The existenceof atoms was first postulatedon philosophical grounds by Ionian scholarsin the 5th century B.C. The conceptwas formalized into scientific theory early in the 19thcentury, owing largely to the work of the chemist Dalton and his contemporaries.The exact structureof atoms was not known, but at that time they were believedto be indivisible. Later in the century (1869), Mendeleevproduced the first periodic table, an ordering of the chemical elements according to the weights of their atoms and arrangementin a grid according to their chemicalproperties. For a time it was believedthat completion of the periodic table would representthe final stepin understandingthe structure of matter. Events of the late 19th and early 20th centuries,beginning with the dis- covery of x rays by Roentgen(1895) and radioactivity by Bequerel (1896), revealedthat atoms had a substructureof their own. In 1910, Rutherford pre- sentedexperimental evidence indicating that atoms consisted of a massive, compact,positively chargedcore, or nucleus,surrounded by a diffuse cloud of relatively light, negativelycharged electrons. This model cameto be known as the nuclear atom. The number of positive chargesin the nucleusis called the atomic number of the nucleus(Z). In the electrically neutral atom, the number of orbital electrons is sufficient to balance exactly the number of positive 4 Physics in Nuclear Medicine charges,Z, in the nucleus.The chemicalproperties of an atom are determined by orbital electrons;therefore the atomic number Z determinesthe chemical elementto which the atom belongs. A listing of chemical elementsand their atomic numbersis given in Appendix A. According to classicaltheory, orbiting electronsshould slowly lose energy and spiral into the nucleus, resulting in atomic "collapse." This is qbviously not what happens.The simple nuclear model therefore neededfurther refi~e- ment. This was provided by Niels Bohr in 1913, who presenteda model that has come to be known as the Bohr atom. In the Bohr atom there is a set of stableelectron orbits or "shells" in which electronscan exist indefinitely without loss of energy. The diametersof these shells are determinedby quantum numbers,which canhave only integervalues (n = 1, 2, 3, . . .). Theinnermost shell (n = 1) is called the K shell, the next the L shell (n = 2), followed by the M shell (n = 3), N shell (n = 4), and so forth. Each shell is actually comprisedof a set of orbits, called substates,which differ slightly from one another.Each shell has 2n - 1 substates,where n is the quantum number of the shell. Thus the K shell has only one substate;the L shell has three substates,labeled L1, LII' LIII; and so forth. Figure 1-1 is a schematicrepresentation of the K and L shells of an atom. The M shell and other higher shells have larger diameters. The Bohr model of the atom was further refined with the statementof the Pauli Exclusion Principle in 1925. According to this principle, no two orbital electronsin an atom can move with exactly the samemotion. Becauseof different possible electron "spin" orientations,more than one electron can exist in each substate(Figure 1-1); however, the number of electronsthat can exist in anyone shell or its substatesis limited. For a shell with quantumnumber n,

THE BOHRATOM

~ :~~ ;;;;==--~:: ~ ~ ij - - " " "~ ((I t ::::~:~:. ',\ '\\\)) ~\ \, ", NUCLEUS i I \\:~ "' -<~' /( Ij ---~t.\..\..' ~/~/ ~",~~ KS .~/ ' ' ::-- ~~-=---~... --~~- L SHELL,n=2 Fig. 1-1. SchelIlatic representation'of the Bohr lIIodel of the atom; n is the quantumnumber of the shell. The K shell has one substateand the L shell has three substates.Each substatehas two electrons. Basic Atomic and NuclearPhysics 5

the maximum number of electrons allowed is 2n2. Thus the K shell (n = 1) is limited to two electrons, the L shell (n = 2) to eight electrons, and so forth. The Bohr model is actually an oversimplification. According to modem theories the orbital electrons do not move in precise circular orbits, but rather in imprecisely defined "regions of space" around the nucleus, sometimes actually passing through the nucleus; however, the Bohr model is quite adequate for the purposes of this text.

2. Electron Binding Energies and Energy Levels In the most stable configuration, orbital electrons occupy the innermost shells of an atom, where they are most "tightly bound" to the nucleus. For example, in carbon, which has six electrons, two electrons (the maximum number allowed) occupy the K shell, and the four remaining electrons are found in the L shell. Electrons can be moved to higher shells or completely removed from the atom, but doing so requires an energy input to overcQme the forces of attraction that "bind" the electron to the nucleus. The energy may be provided, for example, by a particle or a photon striking the atom. The amount of energy required to completely remove an electron from a given shell in an atom is called the binding energy of that shell. It is symbolized by the notation KB for the K shell,t LB for the L shell (LIB' LIIB' LIIIB for the L shell substates), and so forth. Binding energy is greatest for the innermost shell; i.e., KB > LB > MB. . . . Binding energy also increases with the positive charge (atomic number Z) of the nucleus, since a greater positive charge exerts a greater force of attraction on an electron. Therefore binding energies are greatest for the heaviest elements. Values of K shell binding energies for the elements are listed in Appendix A. The energy required to move an electron from an inner to an outer shell is exactly equal to the difference in binding energies between the two shells. Thus the energy required to move an electron from the K shell to the L shell in an atom is KB - LB (with slight differences for different L shell substates). Binding energies and energy differences are sometimes displayed on an energy level diagram. Figure 1-2 shows such a diagram for the K and L shells of the element iodine. The top line represents an electron completely separated from the parent atom ("unbound" or "free" electron). The bottom line represents the most tightly bound electrons, i.e., the K shell. Above this are lines representing substates of the L shell. (The M shell and other outer shell lines would be just above the L shell lines.) The distance from the K shell to the top level represents the K shell binding energy for iodine, i.e., 33.2 ke V. To move a K shell electron to the L shell requires about 33 - 5 = 28 ke V of energy.

tSometirnes the notation Kab is also used. 6 Physics in Nuclear Medicine

"FREE" ELECTRONS

4-5 KeV

L SHELL >- ~ Ct: W Z W ~ 33.2 KeV ~ 0 Z ID

K SHELL Fig. 1-2. Electron energy level diagram for an iodine atom. Vertical axis representsenergy required to remove orbital electrons from different shells (binding energy). Removing an electron from the atom, or going from an inner (e.g., K) to an outer (e.g., L) shell, requires an energy input, whereas an electron moving from an outer to an inner shell results in an emissionof energyfrom the atom.

3. Atomic Emissions When an electron is removed from one of the inner shells of an atom, an electron from an outer shell promptly moves in to fill the vacancy, and energy is released in the process. The energy released when an electron drops from an outer to an inner shell is exactly equal to the difference in binding energies between the two shells. The energy may appear as a photon of electromagnetic radiation (Figure 1-3). Electron binding energy differences have exact chara~- teristic values for different elements; therefore the photon emissions are called characteristic radiation or characteristic x rays. The notation used to identify characteristic x rays from various electron transitions is summarized in Table 1-3. Note that some transitions are not allowed. As an alternative to characteristic x-ray emission, the atom may undergo a process known as the Auger (pronounced oh-zhay') effect. In the Auger effect, an electron from an outer shell again fills the vacancy, but the energy released in the process is transferred to another orbital electron. This electron is then emitted from the atom instead of characteristic radiation. The process is shown schematically in Figure 1-4. The emitted electron is called an Auger electron. Basic Atomic and Nuclear Physics 7

LL CY

~fj~: NUCLEU CHARACTERISTIC X RAY

ELECTRON ORBITS Fig. 1-3. Emission of characteristicx rays occurs when orbital electronsmove from an outer shell to fill an inner-shellvacancy. (K" x-ray emissionillustrated.)

The kinetic energy of an Auger electron is equal to the difference between the binding energy of the shell containing the original vacancy and the sum of the binding energies of the two shells having vacancies at the end. Thus the kinetic energy of the Auger electron emitted in Figure 1-4 is KB - 2LB (ignoring small differences in L-substate energies). Two orbital vacancies exist after the Auger effect occurs. These are filled by electrons from other outer shells, resulting in the emission of additional characteristic x rays or Auger electrons. Whether a particular vacancy will result in emission of a characteristic x ray or an Auger electron is a matter of probabilities. The probability that it will yield characteristic x rays is called the fluorescent yield, symbolized by CJ}Kfor

Table 1-3 SomeNotation Used for CharacteristicX Rays Shell With Shell From Vacancy Which Filled Notation K ~ Not allowed K ~, K"2 K ~n K"t K M, Not allowed K Mn Kjj3 K Mill Kjjl K N, Not allowed K Nn.ill Kjj2 Ln M[V Lp, Lm M[V Lcx2 ~ My L", 8 Physics in Nuclear Medicine

K L~, '-

:{ff::: . ::!:f!:: NUCL " ./ ""-. AUGER ELECTRON ELECTRON ORBITS Fig. 1-4. Emissionof an Auger electronas an alternativeto x-ray emission.No x ray is emitted.

1.0

~ 0.8 3 0 ~ ;:w 0.6

I- z W u ~ 0.4 Q: 0 ~ ~ ~ 0.2

0.0 20 40 60 80 100 ATOMIC NUMBER,Z Fig. 1-5. Fluorescentyield (l)K' or probability that an orbital electron shell vacancy will yield characteristicx rays rather than Auger electrons,versus atomic numberZ of the atom. Basic Atomic and Nuclear Physics 9

the K shell, WLfor the L shell, and so forth. Figure 1-5 is a graph of WKversus Z. Both characteristicx rays and Auger electronsare emitted by all elements, but heavy elements are more likely to emit x rays (large w), whereas light elementsare more likely to emit electrons(small w). The notationused to identify the shellsinvolved in Auger electronemission is eabc'where a identifies the shell with the original vacancy, b the shell from which the electron droppedto fill the vacancy, and c the shell from which the Auger electron was emitted. Thus the electronemitted in Figure 1-4 is a KLL Auger electron, symbolizedby eKu. In the notation eKxx'the symbol x is arbi- trary, referring to all Auger electronsproduced from initial K shell vacancies.

D. THE NUCLEUS

1. Composition The atomic nucleus is comprisedof protons and neutrons. Collectively theseparticles are known as nucleons.The propertiesof nucleonsand electrons are summarizedin Table 1-4. Nucleons are much more massive than electrons (by nearly a factor of 2000). On the other hand, nuclear diametersare very small in comparisonto atomic diameters(10-13 cm versus 10-8 cm). Thus it can be deducedthat the density of nuclear matter is very high (-1014 g/cm3) and that the rest of the atom (electroncloud) is mostly empty space.

2. Terminology and Notation An atomic nucleus is characterized by the number of neutrons and protons it contains. The number of protons determines the atomic number of the atom, Z. As mentioned earlier, this determines also the number of orbital electrons in the electrically neutral atom and therefore the chemical element to which the atom belongs. The total number of nucleons is the mass number of the nucleus, A. The difference, A - Z, is the neutron number, N. The mass number A is approximately equal to, but not the same as the atomic weight (A W) used in chemistry. The latter is the average weight of an atom of an element in its natural abundance (Appendix A).

Table 1-4 Basic Propertiesof Nucleonsand Electrons Particle Charget Mass (u) Mass (MeV) Proton + I 1.0075~3 938.211 Neutron 0 1.008982 939.505 Electron -1 0.000548 0.511 ~ tOne unit of chargeis equivalenlto 1.602 x 10-19coulombs. 10 Physics in Nuclear Medicine

The notation now used to summarize atomic and nuclear composition is :::XN'where X represents the chemical element to which the atom belongs. For example, an atom comprised of 53 protons, 78 neutrons (and thus 131 nucleons), and 53 orbital electrons represents the element iodine and is symbolized by 1~~178.Since all iodine atoms have atomic number 53, either the "I" or the "53" is redundant, and the "53" can be omitted. The neutron number, 78, can be inferred from the difference 131 - 53 and also can be omitted. Therefore a shortened but still complete notation for this atom is 1311.An acceptable alternative in terms of medical terminology is 1-131. Obsolete forms {often found in older texts) include r31, 13J, and 1131.

3. Nuclear Families Nuclear species are sometimes grouped into families having certain common characteristics. A nuclide is characterized by an exact nuclear composition, including the mass number A, atomic number Z, and arrangement of nucleons within the nucleus. To be classified as a nuclide, the species must have a "measurably long" existence, which for current technology means a lifetime greater than about 10-12 sec. For example, 12C, 160, and 1311are nuclides. Nuclides that have the same atomic number Z are called isotopes. Thus 12sl, 121, and 1311are isotopes of the element iodine. Nuclides with the same X 131C N l.d . mass number A are Isobars-e.g.,. 1311, 131 e, s. uc I es WIth the same X 133C A . neutron number N are lsotones-e.g.,. 131S3178' 132S4 e78' SS S78. mnemonIc de- vice for remembering these relationships is that isotopes have the same number of protons, isotones the same number of neutrons, and isobars the same mass number.

4. Nuclear Forces and Energy Levels Nucleons within the nucleus are subject to two kinds of forces. Repulsive coulombic or electrical forces exist between positively charged protons. These are counteracted by very strong forces of attraction, called exchange forces, between any two nucleons. Exchange forces are effective only over very short distances, and their effects are seen only when nucleons are very close together, as they are in the nucleus. Exchange forces hold the nucleus together against the repulsive coulombic forces between protons. Nucleons move about within the nucleus in a very complicated way under the influence of these forces. One model of the nucleus, called the shell model, portrays the nucleons as moving in "orbits" about one another in a manner similar to that of orbital electrons moving about the nucleus in the Bohr atom. Only a limited number of motions are allowed, and these are determined by a set of nuclear quantum numbers. The most stable arrangement of nucleons is called the ground state. Other arrangements of the nucleons fall into two categories: Basic Atomic and Nuclear Physics 11

1. Excited states are arrangementsthat are so unstable that they have only a transientexistence before transforminginto someother state. 2. Metastablestates are also unstable,but they have relatively long lifetimes before transforming into another state. These are also called isomeric states. The dividing line for lifetimes between excited and metastablestates is about 10-12 sec. This is not a long time accordingto everydaystandards, but it is "relatively long" by nuclear standards.Some metastablestates are quite long-lived, i.e., averagelif~times of several hours. (The prefix meta derives from the Greek word for' 'almost.") Becauseof this, metastablestates are consideredto have separateidentities and are classifiedthemselves as nuclides. Two nuclides that differ from one anotherin that one is a metastablestate of the other are called isomers. In nuclear notation, excited statesare identified by an asterisk (AX*) and metastablestates by the letter m (AmXor X -Am). t Thus 99mTc(or Tc-99m) representsa metastablestate of 99Tc,and 99mTcand 99Tcare isomers. Nuclear transitions between different nucleon arrangementsinvolve discrete and exact amountsof energy, as do the rearrangementsof orbital electrons in the Bohr atom. A nuclear energylevel diagram is usedto identify the various excited and metastablestates of a nuclide and the energy relationshipsamong them. Figure 1-6 showsa partial diagramfor 131Xe.:j:The bottom line represents the ground state, and other lines representexcited or metastablestates. Meta- stable statesusually are indicated by somewhatheavier lines. The vertical dis- tancesbetween lines are proportional to the energydifferences between levels. A transition from a lower to a higher state requires an energy input of some sort, e.g., a photon or particle striking the nucleus.Transitions from higher to lower statesresult in the releaseof energy, which is given to emitted particles or photons.

5. Nuclear Emissions The energyreleased in a nucleartransformation to a more stablestate may appearas a photon of electromagneticradiation. Photonsof nuclear origin are called"( rays (gammarays). The energydifference between the statesinvolved in the transition determinesthe "(-ray energy. For example, in Figure 1-6 a transition from the level marked0.364 MeV to the ground statewould produce a 0.364 MeV "( ray. A transition from the 0.364 MeV level to the 0.080 MeV level would producea 0.284 MeV "( ray. As an alternativeto emitting a "( ray, the nucleusmay transfer the energy to an orbital electron and emit the electron insteadof a photon. This process, which is similar to the Auger effect in x-ray emission(Section C.3), is called internal conversion.It is discussedin detail in Chapter2, SectionE. tThe notation AX'"is common in Europe (e.g.. ""Tcm). :l:Actually, theseare the excited and metastablestates fonned during radioactivedecay by 13- emissionof 131(Chapter 2, SectionD). ' 12 Physics in Nuclear Medicine

0.8

0.722 0.7 0.667 0.637

~ 6 > o. Q) ~ >- o.5 ~ a: w z o.4 0.404 w - 0.364 w > ' 0.34 I I- o.3

o. 0.080 (METASTA8LE)

GROUND STATE 0.0 Fig. 1-6. Partial nuclear energy level diagram for 131Xenucleus. Vertical axis representsenergy differences between nuclear states (or "arrangements"). Going up the scale requiresenergy input. Coming down the scaleresults in the emissionof nuclearenergy. Heavierlines indicate metastablestates.

6. Nuclear Binding Energy When the mass of an atom is comparedto the sum of the massesof its individual components(protons, neutrons,and electrons),it is alwaysfound to be less by some amount, Am. This massdeficiency, expressedin energyunits, is called the binding energyEB of the atom:

EB = ~mc2 (1-4)

For example, consider an atom of 12C. This atom is comprised of six protons, six electrons, and six neutrons, and its mass is precisely 12.0 u (by definition of the universal mass unit u). The sum of the massesof its components is

electrons 6 x 0.000055u = 0.000330u 6 x 1.007593u = 6.045558u protons neutrons 6 x 1.008982u = 6.053892u 12.099780 u Basic Atomic and Nuclear Physics 13

Thus ~m = 0.09978 u. Since 1 u = 931.5 MeV, the binding energy of a 12C atom is 0.099780 x 931.5 = 92.95MeV. The binding energyis the minimum amountof energyrequired to overcome the forces holding the atom together in order to separateit completely into its individual components.Some of this representsthe binding energy of orbital electrons,i.e., the energy required to strip the orbital electronsaway from the nucleus;however, comparisonof the total binding energy of a 12Catom with the K shell binding energy of carbon (Appendix A) indicatesthat most of this energyis nuclear binding energy,i.e., energyrequired to separatethe nucleons. Nuclearprocesses that result in the releaseof energy(e.g., -y-rayemission) always increasethe binding energy of the nucleus. Thus a nucleusemitting a 1 MeV -y ray would be found to weigh less (by the massequivalent of 1 MeV) after the -yray was emitted than before. In essence,mass is convertedto energy in the process.

7. Characteristics of Stable Nuclei Not all combinationsof protonsand neutronsproduce stable nuclei. Some

areunstable, even in their groundstates. An unstablenucleus emits particles ~ and/orphotons to transformitself into a more stablenucleus. This is the process of radioactive disintegration or radioactive decay, discussedin Chapter 2. A surveyof the generalcharacteristics of naturally occurring stable nuclidespro- vides clues to the factors that contribute to nuclear instability and thus to radioactive decay. A first observationis that there are favored neutron-to-protonratios among stable nuclides. Figure 1-7 is a plot of the stable nuclides according to their neutronand proton numbers.For example, the nuclide I~C is representedby a dot at the point Z = 6, N = 6. The stable nuclides are clusteredaround an imaginary line called the line of stability. For light elements,the line correspondsto N = Z, i.e., about equal Qumbersof protonsand neutrons.For heavy elements,it correspondsto N = 1.5Z, i.e. about50 percentmore neutronsthan protons. The line of stability ends at 2O9Bi.All heavier nuclides are unstable. In general, there is a tendencytoward instability in systemscomprised of large numbersof identical particles confined in a small volume. This explains the instability of very heavy nuclei. It also explains why, for light elements, stability is favoredby more or lessequal numbers of neutronsand protonsrather thangrossly unequalnumbers. A moderateexcess of neutronsis favoredamong heavier elementsbecause neutrons provide only exchangeforces (attraction), whereasprotons provide both exchangeforces and coulombicforces (repulsion). Exchangeforces are effective over very short distancesand thus affect only "close neighbors" in the nucleus, whereasthe repulsive coulombic forces are effective over much greater distances.Thus an excessof neutronsis required in heavynuclei to overcomethe long-rangerepulsive Coulombic forces between a large numberof protons. Nuclides that are not close to the line of stability are likely to be unstable.

, L, 14 Physics in Nuclear Medicine

14 I ISOTONES

(/) 0-1"<9 &0 00::0

120 0 1'~"i> 0 t " OS' oor 0 I"T1 ro (/) :1 0 .

°ro

100 looro r 0 z 0 oor:0,0 0

a:" 0 ro w 01: m 80 :r-.°oo ~=> 0;M°roo : z oj":"'0 0

0 :.F! ~ 60 l- => w z

'AC

0 20 40 60 80 100 ATOMIC NUMBER,Z Fig. 1-7. Atomic number2 versusneutron number N for the stablenuclides. They are I. clusteredaround an imaginary line called the line of stability, N = 2 for light elements; ! N = 1.52 for heavy elements. f

Unstablenuclides lying above the line of stability are said to be "proton deficient," while those lying below the line are "neutron deficient." Unstable nuclidesgenerally undergo radioactive decay processes that transformthem into nuclideslying closer to the line of stability, as discussedin Chapter2. From Figure 1-7 it can be seenthat there are often many stable isotopes of an element.Isotopes fall on vertical lines in the diagram.For example,there are ten stableisotopes of tin (Sn,t Z = 50). There may also be severalstable

tWhile most element symbols are simply one- or two-letter abbreviationsof their (English) names, II symbols derive from Latin or Greek namesof metals known for over two millennia: antimony(stibium, Sb); copper(cuprum, Cu); gold (aurum, Au); iron (ferrum, Fe); lead (plumbum, Pb); mercury (hydrargyrum,Hg); potassium(kalium, K); silver (argentum,Ag); sodium (natrium, Na); tin (stannum,Sn); tnngsten(wolfram, W). Basic Atomic and Nuclear Physics 15

isotones. These fall along horizontal lines. In relatively few cases, however, is there more than one stable isobar (isobars fall along descending 45° lines on the graph), reflecting the existence of several modes of "isobaric" radioactive decay that permit nuclides to transform along isobaric lines until the most stable isobar is reached. This will be discussed in detail in Chapter 2. One also notes among the stable nuclides a tendency to favor even numbers. For example, there are 165 stable nuclides with both even numbers of protons and even numbers of neutrons. Examples are ~He and l~C. There are 109 "even-odd" nuclides, with even numbers of protons and odd numbers of neutrons or vice versa. Examples are ~Be and l;B. However, there are only four stable "odd-odd" nuclides: ~H, ~i, l~B, and l~N. The stability of even numbers reflects the tendency of nuclei to achieve stable arrangements by the' 'pairing up" of nucleons in the nucleus. Another measure of relative nuclear stability is nuclear binding energy, since this represents the amount of energy required to break the nucleus up into its separate components. Obviously, the greater the number of nucleons, the greater the total binding energy. Therefore a more meaningful parameter is the binding energy per nucleon, EB/A. Higher values of EB/A are indicators of greater nuclear stability. Figure 1-8 is a graph of EB/A versus A for the stable nuclides. Binding

c:r ~ 9 > OJ ~ 8 a3 W Z 7 0 W -J 6 u ::> Z 5 Q:: W 0- 4 >- ~ ffi 3 Z w 2 ~ Z ;:; I Z a3 0 20 40 60 80 100 120 140 160 180 200 220 240 l Fig. 1-8. Binding energy per nucleonMASS versus NUMBER,A mass number for the stable nuclides.

[Adapted with permission from Evans RD, The Atomic Nucleus. New York, McGraw- Hill, 1972, p. 299].

I. 16 Physics in Nuclear Medicine energy is greatesi(~8 MeV/nucleon) for nuclides of massnumber A = 60. It decreases slowly with increasingA, indicating the tendencytoward instability for very heavy nuclides. Finally, there are a few peaks in the curve representing very stable light nuclides, including iRe, I~C, and I~O. Note that these are all "even-even" nuclides.

REFERENCES

Basic mathematicsand physics are reviewed in many textbooks, including the following: Boyd CM, Dalrymple GV: Basic SciencePrinciples of Nuclear Medicine. St. Louis, C. V. Mosby, 1974 Kemp LAW: Mathematicsfor Radiographers.Philadelphia, F. A. Davis, 1964 Kemp LAW, Oliver R: Basic Physicsin Radiology (ed 2). Oxford, Blackwell Scientific, 1970 Nave CR, Nave BC: Physics for the Health Sciences.Philadelphia, W. B. Saunders, 1975 Recommendedtexts for in-depthdiscussions of topics in atomic and nuclearphysics are the following: EvansRD: The Atomic Nucleus. New York, McGraw-Hill, 1972 Lapp RE, Andrews HL: Nuclear Radiation Physics (ed 4). Englewood Cliffs, N.J., Prentice-Hall, 1972 Rollo FD: Atomic and nuclear physics, in Rollo FD (ed): Nuclear Medicine--Physics, Instrumentation,and Agents. St. Louis, C. V. Mosby, 1977, chap 1